1 Introduction

The optimization process involves selecting the optimal solution from a group of different available solutions (Abdel-Basset et al. 2018). In the optimization field, a lot of traditional and non-traditional techniques had been proposed and applied to select the optimal solution (Razmjooy et al. 2021a). The traditional methods usually are non-robust and time-consuming. A class of non-traditional techniques namely population-based algorithms emerged due to the wide availability of high computational efficiency (Sangaiah et al. 2020). A population-based algorithm is a metaheuristic algorithm that mimics the social behavior of different groups such as insects, animals, birds, etc. which are found in nature. The population-based algorithms have seen an increase in their use in real optimization issues due to their adaptability, flexibility, effectiveness, simplicity, and excellent global searching ability (Wu et al. 2019). Some examples of population-based algorithms are Particle Swarm Optimization algorithm (PSO)(Kennedy and Eberhart 1995), Ant Colony Optimization algorithm (ACO) (Dorigo et al. 2006), Artificial Bee Colony algorithm (ABC) (Karaboga and Basturk 2007), Bat Algorithm (BA) (Yang 2010), Grey Wolf Optimization (GWO) (Mirjalili et al. 2014), Moth-Flame Optimization (MFO) (Mirjalili 2015), Whale Optimization Algorithm (WOA) (Mirjalili and Lewis 2016), Salp Swarm Algorithm (SSA) (Mirjalili et al. 2017), Harris Hawks Optimization (HHO) (Heidari et al. 2019), and Grasshopper Optimization Algorithm (GOA) (Saremi et al. 2017). Figure 1 shows a taxonomy of the metaheuristic optimization algorithms.

Fig. 1
figure 1

Taxonomy of Metaheuristic Optimization Algorithms

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In the latest years, the Grasshopper Optimization Algorithm (GOA) has been broadly studied and applied in numerous fields due to the following reasons: (i) its simple implementation, (ii) reasonably good optimization capability, and (iii) relatively incredible overall performance in realizing complicated troubles. GOA had been proposed to solve various optimization problems in many domains in previously published works, such as constrained and unconstrained test functions (Neve et al. 2017; Saremi et al. 2017), knapsack problem (Pinto et al. 2019), task assignment problem(L. Xu et al. 2020a, b), hand posture estimation problem (Saremi et al. 2020), power management (Juhari et al. 2019; Jumani et al. 2019; Rajput et al. 2017; Talaat et al. 2020), electric load scheduling in smart grids (Jamil and Mittal 2020; Ullah et al. 2019, 2020), hydrothermal scheduling (Zeng et al. 2021).

Although the GOA has various advantages as mentioned, it has some drawbacks in that the performance of GOA in its original version has slow speed convergence and a high probability to get a local solution instead of a global solution. For that, the improvement of the performance of GOA is an important issue to address its deficiencies.

In the current study, novel enhancements for GOA which are named NVGOA variants have been proposed to improve both the exploitation and the exploration phases of the GOA. The proposed enhancements’ performance has been compared with the original version of GOA for confirmation of the effectiveness of the two proposed mechanisms. Moreover, a comparative analysis has been accomplished with other six state-of-the-art optimization techniques as well as with other enhancements of GOA for verification of the efficiency and feasibility of the proposed enhancements. Also, for reality evaluation, the proposed NVGOA variants have been applied to schedule the load for the reduction of the operating cost and the peak in the smart grid operation. In the next sub-section, all the contributions of this study have been mentioned.

1.1 The work contributions

1.1.1 The contributions of the proposed algorithms can be summarized as:

  1. 1.

    Eight enhancements called: Novel variants of GOA (NVGOAs) are proposed based on two strategies: a grouping mechanism of non-linear comfort zone parameters which groups and a mutation mechanism.

  2. 2.

    Two different groups of non-linear ‘c’ parameters are proposed.

  3. 3.

    Extensive simulation experiments are conducted on CEC 2017 benchmark functions to study the impact of each mechanism alone and combined.

  4. 4.

    Several comparison procedures on CEC 2017 benchmark functions have been accomplished with other optimization techniques such as original GOA, PSO, MFO, SSA, HHO, WOA, and GWO; as well as, with some previous enhancements of GOA based on mean and standard deviation for validation of the proposed NVGOA variants.

  5. 5.

    Statistical analysis using the Wilcoxon Signed-Rank test is conducted to assure the effectiveness of the proposed variants.

  6. 6.

    Application of the proposed NVGOA algorithms for load scheduling at demand side management (DSM) to decrease the peak demand and the electricity cost in three different regions: residential, commercial, and industrial regions.

The rest of this paper is structured as follows: Sect. 2 mentions the related works. Section 3 provides an overview of the original Grasshopper Optimization Algorithm. In Sect. 4, the novel variants of GOA (NVGOAs) are presented. Section 5 discusses the extensive simulation results of the application of the proposed NVGOAs variants on CEC-2017 benchmark functions and DSM. Finally, a brief conclusion is provided in Sect. 6.

2 Related works

In (Mafarja et al. 2019), Mafarja et al. proposed a binary GOA (BGOA) variant by applying it to the feature selection problem. In similar work (Hichem et al. 2022), Hichem et al. solved the feature selection problem by proposing another BGOA enhancement called NBGOA and compared the performance of NBGOA to five optimization algorithms. Twenty datasets of various sizes were used to test the performance of all these algorithms. The results demonstrated that the proposed variant could outperform the other optimizers. In (Algamal et al. 2020), Algamal et al. proposed an enhancement in BGOA based on four quadratic transfer functions which were adapted to improve the performance of the BGOA in quality of the quantitative structure–activity relationship (QSAR) modeling of the H1N1 virus.

Chaos theory was integrated into GOA by Arora and Anand in (Arora and Anand 2019) to increase the speed of convergence to the optimal solution and applied this modification to the 30 most widely benchmark functions. In (Saxena 2019), Saxena proposed a crossover operator besides the chaos theory for solving the CEC-2017 benchmark functions, and some real problems such as prediction of protein structure (PSP), model order reduction (MOR), and sound wave parameter synthesis problem with frequency modulation. In (Z. Xu et al. 2020a, b), Xu et al. imported chaos theory in addition to orthogonal learning into GOA, named it OLCGOA, and used it solve the CEC-2017 benchmark functions and the feature selection problem. The results showed that OLCGOA had a more appropriate solution and stronger global optimization ability.

Saxena and Kumar (Saxena and Kumar 2020) integrated chaos theory into GOA and called this variant Enhanced Chaotic Grasshopper Optimization Algorithm (ECGOA). The proposed ECGOA algorithm was developed to solve protein structure prediction problems by applying it to artificial and real protein sequences.

Wang et al. (Wang et al. 2021) proposed a chaotic map to integrate into GOA in addition to another modification in the 'c' coefficient and then applied it to solve the CEC-2017 benchmark functions and some real optimization problems such as the design of pressure vessels, the design of multiple disk clutch brake, and I-beam design.

Wenhan et al. (Wenhan et al. 2019) imported chaos theory into GOA in enhancement named CGOA to reduce the energy consumption of chiller loading and tested the proposed variant on 4 standard benchmark functions. The simulation results demonstrated that the power consumption was decreased from 90 to 40% using CGOA.

Luo et al. (Luo et al. 2018) suggested an improved version of the GOA which mixes gaussian mutation, levy-flight, and opposition-based learning; to solve the financial stress forecast problem and continuous optimization problems.

Algamal et al. (Algamal et al. 2021) imported the gaussian kernel into GOA to solve feature selection and support vector machine (SVM) parameter selection.

Zhao et al. (Zhao et al. 2019) introduced another enhancement based on integrating random jumping and dynamic weight strategy. Zhou et al. (Zhou et al. 2020) introduced three strategies; Cauchy mutation, genetic mutation, and orthogonal learning; to integrate into GOA and tested them on CEC2017 functions. Ewees et al. (Ewees et al. 2018) introduced an enhanced version of GOA by incorporating opposition-based learning with GOA to tackle 23 benchmark functions and some design problems in the engineering field such as the design of welded beams, pressure vessels, tension/compression springs, and three-bar truss.

Raeesi et al. (Raeesi et al. 2020) integrated the opposition-based learning into GOA and applied it to tackle unimodal and multimodal benchmark functions and tuning parameters problems of the Takagi–Sugeno-Kang (TSK) model.

Bairathi and Gopalani (Bairathi and Gopalani 2020) also incorporated opposition-based learning into GOA and tested its performance using 25 well-known benchmark functions. Steczek et al. (Steczek et al. 2020) imported four techniques; grey wolf optimizer, dynamic adaptation of the ‘c’ parameter, natural selection, and opposite-based learning; to GOA for selective harmonic elimination problems in inverters.

Taher et al. (Taher et al. 2019) proposed an additional mutation process to GOA to address optimal power flow problems, and testing the proposed model using modification was suggested by standard IEEE 30-bus, 57-bus, and 118-bus systems.

In (X. Yue and Zhang 2020), Yue and Zhang imported a strategy based on Principal Component Analysis (PCA) and novel inertia weight into GOA. Feng et al. (Feng et al. 2020) incorporated three strategies; \(\beta\) hill-climbing, niche mechanism, and nonlinear ‘c’ parameter; into GOA for tackling the bin packing problem.

In (Tanwar et al. 2020), Tanwar et al. integrated fractional calculus into GOA to control sensor activation in wireless sensor networks.

Mishra et al. (Mishra et al. 2020) proposed a random walk theory to improve the GOA performance and achieve a fine balance between exploitation and exploration areas.

In (Bala et al. 2020), Bala et al. suggested a hybrid of an Echo State Network (ESN) strategy with GOA for the prediction of airplane engine faults.

Goel et al. (Goel et al. 2020) combined a random forest approach with GOA for the detection of the autism spectrum disorder problem. The results demonstrated that autism spectrum disorder can be properly identified in children, adolescents, and adults. Great accomplishments have been made: Accuracy rates for the datasets for children with autism were 100%, adolescents with autism were 100%, and adults with autism were 99.29%.

Huang et al. (Huang et al. 2020) imported a non-linear ‘c’ parameter, exemplar pool mechanism, and social interaction mechanism; into GOA to optimize the parameters of the hybrid active power filter.

In (El-Shorbagy and Ayoub 2021), El-Shorbagy and Ayoub integrated a local search (LS) strategy into GOA to solve the data clustering problem. The experimental results and statistical analysis confirmed that the combination between GOA and LS outperformed mentioned techniques in the work.

Other variants of GOA were suggested in previous works by hybridization of some well-known algorithms with GOA to speed the GOA and improve its performance. Examples of some algorithms which proposed to hybrid with GOA such as; ABC algorithm (Sharma et al. 2021), Bat Algorithm (S. Yue and Zhang 2021), and Gravity Search Algorithm (GSA)(Guo et al. 2020) to solve numerical problems. Other studies proposed hybrid GOA with Genetic Algorithm (GA) for optimizing the non-linear equations system (El-Shorbagy and El-Refaey 2020), with Cuckoo Search (CS) algorithm for optimizing the day-ahead scheduling of microgrid and its optimal operation (C et al. 2022), with Differential Evolution algorithm (DE) for tackling the classification of colors of dyed fabrics(Li et al. 2021), with Cat Swarm Optimization algorithm (CSO) for feature section problem and optimizing the multi-layer perceptron design (Bansal et al. 2020), and with Artificial Neural Network (ANN) to optimize the skin color detection (Razmjooy et al. 2021b). Table 1 shows the summary of the mentioned works.

Table 1 Summary of GOA-Enhancements
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3 The grasshopper optimization algorithm in brief

One of the recent population-based and bio-inspired algorithms is a Grasshopper Optimization Algorithm (GOA) that simulates grasshopper swarms’ behavior (Saremi et al. 2017). GOA was introduced in 2017 by Saremi et al. Grasshoppers are insects that may be considered a pest under some circumstances. The life cycle of a grasshopper comprises two phases: nymph and adult. The movement of the nymphs is gradual with small steps and slow movement in comparison to the adults' which is broad, long-range, and abrupt. Figure 2 shows the real grasshopper, the life cycle of the grasshopper, and the grasshoppers’ swarm.

Fig. 2
figure 2

a Real grasshopper, b Grasshopper life cycle, (c) Grasshoppers swarm

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The exploration and exploitation of the search area are two essential phases of optimization. The grasshoppers provide these two phases throughout the food search via social interactions. GOA can solve real issues with unknown search areas.

The grasshopper positions in the swarm are affected by three evolutionary forces: a social interaction force, a gravity force, and a wind force, to update the position of individuals in the swarms. The mathematical model used to simulate grasshoppers’ swarm behavior is presented using Eq. (1)

$${\text{X}}_{{\text{i}}} \, = \,{\text{S}}_{{\text{i}}} \, + \,{\text{G}}_{{\text{i}}} \, + \,{\text{W}}_{{\text{i}}}$$
(1)

where \({X}_{i}\) represents the \({i}^{th}\) grasshopper position, \({S}_{i}\) is the social interaction force, \({G}_{i}\) is the force of gravity on the \({i}^{th}\) grasshopper, and \({W}_{i}\) is the wind force.

The \(S\) component in Eq. (1) is calculated as follows:

$${\text{S}}_{{\text{i}}} \, = \,\sum\limits_{{{\text{j}}\, = \,1,\,{\text{j}}\, \ne \,{1}}}^{{\text{N}}} {{\text{s}}\,\left( {{\text{d}}_{{{\text{ij}}}} } \right)} \,{\text{d}}_{{{\text{ij}}}}$$
(2)

where, N is the number of grasshoppers. \({d}_{ij}\) is the distance between the \({i}^{th}\) and the \({j}^{th}\) grasshoppers and it is calculated as \({d}_{ij}=\left|{x}_{j}-{x}_{i}\right|\). \({\widehat{d}}_{ij}=\frac{{x}_{j}-{x}_{i}}{{d}_{ij}}\) is a unit vector from the \({i}^{th}\) and the \({j}^{th}\) grasshopper. s is a function to define the strength of social forces and can be given by Eq. (3):

$${\text{s}}\left( {{\text{d}}_{{{\text{ij}}}} } \right)\, = \,fe^{{\left( {\frac{{ - {\text{d}}_{{{\text{ij}}}} }}{{\text{l}}}} \right)}} \, - \,{\text{e}}^{{ - {\text{d}}_{{{\text{ij}}}} }}$$
(3)

where, f indicates the attraction intensity, and l is a scale of the attractive length. According to (Saremi et al. 2017), the best-chosen values of f and l are equal to 0.5 and 1.5 respectively.

The G component in Eq. (1) is calculated as follows:

$$G_{{\text{i}}} \, = \, - g\hat{p}_{{\text{g}}}$$
(4)

where, \(g\) is the gravitational constant, and \({\widehat{p}}_{g}\) shows a unity vector towards the earth's center.

Also, the W component in Eq. (1) can be calculated by Eq. (5):

$${\text{W}}_{{\text{i}}} \, = \,w\hat{p}_{{\text{w}}}$$
(5)

where, w is a constant drift, and \({\widehat{p}}_{w}\) is a unit vector in the direction of the wind.

By replacing S, G, and W in Eq. (1) with Eqs. (2,4,5), Eq. (1) becomes as follows:

$${\text{X}}_{{\text{i}}} \, = \,\sum\limits_{{{\text{j}}\,{ = }\,{1,}\,{\text{j}}\, \ne \,1\,}}^{{\text{N}}} {{\text{s}}\,\left( {\left| {x_{{\text{j}}} - x_{{\text{i}}} } \right|} \right)\,\frac{{x_{{_{j} }} \, - \,x_{{\text{i}}} }}{{{\text{d}}_{{{\text{ij}}}} }}\, - g\hat{p}_{g} \, + \,w\hat{p}_{w} }$$
(6)

However, the previous mathematical model cannot be used directly to solve optimization problems. For that, a modified version from Eq. (6) was presented by (Saremi et al. 2017) to resolve optimization problems as follows:

$${\text{X}}_{{\text{i}}}^{{\text{d}}} \, = \,{\text{c}}\,\left( {\sum\limits_{{{\text{j}}\,{ = }\,{1,}\,{\text{j}}\, \ne \,{1}}}^{N} {{\text{c}}\,\frac{ub\, - \,lb}{2}\,{\text{s}}\left( {\left| {x_{{\text{j}}}^{{\text{d}}} \, - x_{{\text{i}}}^{{\text{d}}} } \right|} \right)\,\frac{{x_{{\text{j}}} \, - \,x_{{\text{i}}} }}{{{\text{d}}_{{{\text{ij}}}} }}\,} } \right)\, + \,\hat{T}_{{\text{d}}} \,$$
(7)

where, ub is the upper bound, and lb is the lower bound. d is the dimension. c is a comfort zone parameter that affects the grasshopper to move from the repulsion zone or attraction zone into the comfort zone. The gravity is not considered in Eq. (7) (no G component) and the wind direction (W component) is assumed always towards a target solution \(\widehat{T}\). Equation (7) is a final form that describes the position of each grasshopper in a swarm and the behavior of the swarm mathematically. The objective of this equation is the grasshopper's swarm access to its target at the end with a consideration of the interspaces between each grasshopper and the other. In other words, the objective of Eq. (7) is to reduce the term \(c\left({\sum }_{j=1,j\ne i}^{N}c\frac{ub-lb}{2}s\left(\left|{x}_{j}^{d}-{x}_{i}^{d}\right|\right)\frac{{x}_{j}-{x}_{i}}{{d}_{ij}}\right)\) until the position of ith grasshopper (\({X}_{i}\)) in the swarm is equal to its position on the food or its target (\(\widehat{T}\)).

For achieving the balance of exploration and exploitation and for tuning the comfort zone, ‘c’ is required to be decreased proportionally to the iteration number as in Eq. (8):

$$c_{org} = c_{max} - j\frac{{c_{max} - c_{min} }}{J}$$
(8)

where, \({c}_{min}\) is the minimum value of c, and \({c}_{max}\) is the maximum value. j is the current iteration, and J indicates the maximum number of iterations. It is important to mention that the ‘c’ parameter value will always start at unity and decrease to very low values as the iteration number increases. Algorithm 1 shows the pseudocode of the original GOA (Saremi et al. 2017).

4 Novel-Variants of GOA (NVGOA)

The exploration and exploitation processes in the grasshopper optimization algorithm depend on the ‘c’ parameter, which has a significant role in the position update process. However, there are some downsides to the original form of the GOA, such as the probability of stuck in local optima and the slow convergence speed.

Exploration and exploitation of the search space must be balanced to avoid early convergence towards local optima. In this section, novel variants of GOA have been developed to overcome these problems. The ‘c’ parameter has an important role due to the following reasons:

  1. 1.

    The 'c' parameter acts as a link between the exploitation and exploration stages. As a result, even a simple change in the ‘c’ parameter will alter the algorithm’s performance.

  2. 2.

    When the ‘c’ parameter is changed, these areas, which are comfort zone, attraction zone, and repulsion zone, will be affected. By decreasing these zones, it manages the exploratory phase in a way.

figure a

In the original version of GOA, the ‘c’ parameter employed a linear decrease to limit the search space and force all search agents to go toward the target area. The linear decrease in the 'c' parameter was unable to increase the influence of the two stages of the search process: exploitation, and exploration. During the exploitation phase, the parameters frequently caused the search agents to stray from the optimal position, as though they were moving at excessive speeds. While throughout the exploration phase, GOA was unable to swiftly converge around the target, and the search agents simply meandered over the whole search space, providing no strong foundation for the later search phase. The algorithm could not fully utilize the whole iterations due to the linear decreasing parameter mechanism. This behavior of GOA raises the probability of becoming stuck in a local solution, which should be avoided. Two modifications have been suggested in the current study to address these issues. The proposed modifications have been assumed to speed up the GOA convergence and avoid premature convergence.

These modifications are:

  1. 1.

    Modifying the ‘c’ parameter equation based on a grouping mechanism and in non-linear form.

  2. 2.

    Adding simple position mutation step after updating position step.

4.1 The first proposed modification: the grouping mechanism

The proposed grouping mechanism has three key important parameters: the number of ‘c’ parameter equations, the forms of ‘c’ parameter equations, and the method of division of the whole population into sub-population groups.

4.1.1 The number and form of ‘c’ parameter equations

In this work, two groups of ‘c’ parameters have been proposed. Each group consists of four different equations to calculate the ′c′parameter of GOA.

I) The first proposed group of ′c′ parameters In this set, some equations of coefficient 'c' which were motivated by (Huang et al. 2020; Liu et al. 2018; Mishra et al. 2020; Saremi et al. 2017)have been grouped in one set to act as one ‘c’ parameter. The group mechanism for these equations has been proposed as follows:

$$c_{{1_{{\left( {g1} \right)}} }} = c_{org}$$
(9)
$$c_{{2_{{\left( {g1} \right)}} }} = c_{max} - \left( {c_{max} - c_{min} } \right) \times \left( \frac{j}{J} \right)^{2}$$
(10)
$$c_{{3_{{\left( {g1} \right)}} }} = c_{max} - \left( {c_{max} - c_{min} } \right) \times \left( \frac{j}{J} \right)^{\frac{1}{j}}$$
(11)
$$c_{{4_{{\left( {g1} \right)}} }} = e^{{ - 0.5\left( {\frac{q \times j}{J}} \right)^{2} }}$$
(12)

where, \({c}_{{1}_{(g1)}}\), \({c}_{{2}_{(g1)}}\),\({c}_{{3}_{(g1)}}\), and \({c}_{{4}_{(g1)}}\) are four different forms of ‘c’ parameter. \({c}_{{1}_{(g1)}}\) is \({c}_{org}\) from Eq. (8). \({c}_{{2}_{(g1)}}\) is an arc form and it is a non-linear form (NLF). \({c}_{{3}_{(g1)}}\) is an exponential form of NLF. \({c}_{{4}_{(g1)}}\) is another exponential form of NLF. \(q\) is a number to trade-off between exploitation or exploration phases.

The exploitation procedure will be largely supported by high values of q (higher than 6). For q, a reasonable number between three and four can be used.

In the current study,\({c}_{max}\) and \({c}_{min}\) are equal to 1 and 0.00004 respectively. Where, j is the current iteration number, and J is the maximum iteration number.

The idea of grouping the four forms of the ‘c’ parameter is motivated by the non-linear chaotic effect. Figure 3 shows the effect of the four forms of 'c' parameter equations that have been grouped in the first set individually as well as in the group.

Fig. 3
figure 3

The ‘c’ parameters’ values versus iterations individually and after applying the grouping mechanism (set 1)

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II) The second proposed group of ′c′ parameters This is another proposed set of equations of ‘\(c\)’ parameter. Each equation in this set has been multiplied by \({c}_{m}\) Eq. (13) to slightly reduce the clear chaotic effect in the first proposed set.

$$c_{m} = c_{{2\left( {g1} \right)}}$$
(13)
$$c_{{1_{{\left( {g2} \right)}} }} = c_{m} \times \left( {c_{max} - j\frac{{c_{max} - c_{min} }}{J}} \right)$$
(14)
$$c_{{2_{{\left( {g2} \right)}} }} = \left( {c_{m} } \right)^{2}$$
(15)
$$c_{{3_{{\left( {g2} \right)}} }} = c_{m} \times \left( {\left( {cos \left( {\pi \times \frac{j}{J}} \right) + c_{max} } \right) \times \frac{{c_{max} + c_{min} }}{2}} \right)$$
(16)
$$c_{{4_{{\left( {g2} \right)}} }} = c_{m} \times e^{{ - 0.5\left( {\frac{q \times j}{J}} \right)^{2} }}$$
(17)

\({c}_{m}\) is an arc form of NLF. As shown, \({c}_{{1}_{(g2)}}\), \({c}_{{2}_{(g2)}}\), \({c}_{{3}_{(g2)}}\), and \({c}_{{4}_{(g2)}}\) formalize another group consisting of four forms of c parameter equations. \({c}_{{1}_{(g2)}}\) in contrast to \({c}_{{1}_{(g1)}}\) becomes arc adoption of NLF. As well as \({c}_{{2}_{(g2)}}\) which becomes another form of NLF different from its counterpart \({c}_{{2}_{(g1)}}\). \({c}_{{3}_{(g2)}}\) and \({c}_{{4}_{(g2)}}\) also have other forms of NLF. The effect of the four forms of 'c' parameters in the second group individually as well as in the group is shown in Fig. 4.

Fig. 4
figure 4

The ‘c’ parameters’ values versus iterations individually and after applying the grouping mechanism (set 2)

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figure b

4.1.2 The technique of division of the whole population into sub-population groups

There are several methods to divide the population into sub groups, such as random, dynamic, or average division. In the current study, the random division method has been adopted. The equation of the random division method is:

$$c_{r} = 1 + r_{1} \left( {{\text{no}}{.}\, {\text{of}}\, {\text{c}}\, {\text{parameter}}\, {\text{Eqs}}. - 1} \right)$$
(18)

where, \({c}_{r}\) is the equation number of \(c\) parameter from the set of equations based on a random number. The maximum number of \(c\) parameters Eqs. equal to 4 in the current study, and \({r}_{1}\) is a random number between [0,1].

4.2 The second proposed modification: the mutation mechanism

Sometimes, one or more grasshoppers in a swarm may suddenly change their positions for exploration purposes, but this happens with little probability. This is what has been called in this study the mutation mechanism. To improve the performance of GOA, a simple mutation position has been considered randomly with a small probability that is greater than 0 and doesn’t exceed 0.5.

The mutation probability is calculated using Eq. (19):

$$M_{occ} = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{if }}r_{2} \le P_{m} } \hfill \\ 0 \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$$
(19)

where, \({r}_{2}\) is a random number between [0,1], and \({P}_{m}\) is a constant in the range [0,0.5]. \({M}_{occ}\) is a boolean value to express the mutation occurrence, based on \({P}_{m}\). In the state of \({M}_{occ}\) = 1, the mutation strategy has been accomplished according to Eq. (20):

$$x_{im} = lb_{d} + r_{3} \left( {ub_{d} - lb_{d} } \right)$$
(20)

where \({x}_{im}\) is a new position after a mutation occurrence. \(ub\) is the upper bound, \(lb\) is the lower bound, \(d\) is the dimension, and \({r}_{3}\) is a random number between [0,1]. Algorithm 2 shows the pseudocode of the proposed NVGOA variants.

In the current study paper, the impact of each group of ‘c’ parameters without the mutation mechanism has been studied in two proposed variants named NVGOA1_1 and NVGOA2_1 respectively.

Moreover, the impact of the mutation mechanism with the linear’c’ parameter shown in Eq. (8) has been studied with Pm = 0.3 and Pm = 0.5 in two variants called NVGOA3 and NVGOA4 respectively. Finally, the effect of the combination of the two proposed mechanisms has been investigated in NVGOA1_2, NVGOA1_3, NVGOA2_2, and NVGOA2_3. Table 2 shows the details of proposed NVGOAs with the two proposed mechanisms, and Table 3 summarizes all the parameters that must be set in each proposed variant of the NVGOA. The summary of the grasshoppers’ swarm movement to the target by the proposed NVGOA algorithm is shown in Fig. 5.

Table 2 Details of Various GOAs with the two proposed mechanisms
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Table 3 Parameter’s settings of the proposed NVGOA variants
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Fig. 5
figure 5

Summary of the proposed NVGOA algorithm

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5 The simulation results and discussion

5.1 CEC-2017 benchmark functions

For evaluating the performance of the proposed variants, a standard set of mathematical functions are required. This paper has used 29 CEC 2017 benchmark functions (F2 is a dummy) to compare the proposed NVGOA variants' performance with the original GOA, other state-of-the-art optimizers, and other enhancements of GOA. The details of the CEC 2017 benchmark functions such as the search range, composition category, and optima values are shown in Table 4. Different types of benchmark functions can comprehensively estimate the performance of the proposed variants.

Table 4 Details of CEC-2017 benchmark functions
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In the next subsection, the 29 functions’ results of the proposed NVGOA variants are presented. To make the comparison fair, all experiments have been carried out for 30 dimensions and 30 search agents for 1,000 iterations. For ensuring more fairness in the comparison, the initial populations are the same for all the proposed variants, original GOA, and other optimizers. Furthermore, each benchmark function has been performed with 30 independent runs. All experiments have been done using a computer with Ryzen 7 processor and 16 GB RAM and by using MATLAB R2018b.

5.2 Simulation results of CEC-2017 benchmark suite

5.2.1 Case 1: mechanisms comparison experiment

To compare the effects of incorporating the two proposed mechanisms into GOA, the mechanisms comparison experiment has been performed and tested using the CEC-2017 functions. Table 5 records the results of that comparison based on mean, standard deviation (SD), rank, and overall rank metrics. In Table 5, a positive implication of the combined two proposed mechanisms on the performance of GOA can be observed. By looking at the results of the 29 functions, it seems that the original GOA results are close to the results of NVGOA1_1 and NVGOA2_1. However, the results of the other 6 proposed variants are better than GOA for all 29 functions. The best values for each benchmark are shown in bold in the table.

Table 5 Simulation results of CEC-2017 benchmark functions (mechanisms comparison case)
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The ‘count’ row in Table 5 is shown the number of functions in which the variant has recorded the best results while the ‘Avg. rank’ row is shown the average of ranks calculated based on the variants’ rank in each function. Also, the last row in Table 5 shows the overall rank of all GOA variants. By clear observation of the results recorded in Table 5, it is worth mentioning that the mean values are the best for the proposed NVGOA1_2 algorithm in 4 functions while the proposed NVGOA1_3 algorithm has the best results in 6 functions compared to all proposed variants and original GOA. Moreover, the three proposed variants; NVGOA2_2, NVGOA3, and NVGOA4; have recorded the best results in mean for 8 functions. Furthermore, the NVGOA2_3 variant has outperformed in 11 functions in comparison to all proposed variants, so it has got the first overall rank among all the proposed variants and also the original GOA.

The ‘‘+/-/=’’ row in Table 5 refers to whether the performance of the NVGOA2_3, as the best among the proposed algorithms, is better than, worse than, or equal to other variants compared. For instance, the performance of the proposed NVGOA2_2 has been equal to the performance of the NVGOA2_3 algorithm in 3 functions while the NVGOA2_2 variant outperformed NVGOA2_3 in 9 functions. As well, the proposed NVGOA1_3 outperformed the NVGOA2_3 algorithm in 10 functions while the NVGOA2_3 and NVGOA1_3 algorithms' performance has been the same in 2 functions. Additionally, the NVGOA4 and the NVGOA2_3 has been the same results in one function (F30) while the NVGOA2_3 has outperformed the NVGOA4 in 15 functions.

Figure 6 shows the convergence curves of 12 functions (for illustration). Obviously from Fig. 6, the convergence speed of the NVGOA1_3 in most curves has been greater than the proposed NVGOA2_3 and NVGOA4 algorithms however these variants have reached better solutions in most curves than the proposed NVGOA1_3. Moreover, the convergence of other proposed variants has better than the original GOA. Accordingly, the combination of the grouping mechanism (set 1) and the mutation mechanism has sped up the convergence when compared to the second group along with the mutation mechanism which obtains a better solution but slowly in comparison to first one. Generally, the two proposed techniques, the grouping of nonlinear 'c' parameters and mutation, have a great impact to improve GOA performance, speed up the convergence, and help to address the issue of the local solutions.

Fig. 6
figure 6

Convergence curves of the proposed NVGOA variants against the original GOA in case 1 (12 functions)

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5.2.2 Statistical analysis using wilcoxon signed-rank test for case 1

Empirically, the superiority of the proposed variants cannot be confirmed when evaluating the mean and SD values of 30 runs only unless the performance of each run is compared separately. For this purpose, many approaches are employed nonparametric statistical tests such as the Wilcoxon test to know whether the significant differences between correlated samples or not. In the current study, the Wilcoxon signed-rank test has been employed to investigate the significant differences between the proposed variants of GOA and the original GOA.

Table 6 shows the p-values of the Wilcoxon signed-rank test. It can be observed that the p-values of the proposed two variants; NVGOA3 and NVGOA4; in all functions are less than 0.05, indicating that these variants have significantly different from the original GOA. Additionally, the p-values of the proposed NVGOA1_3 and NVGOA2_3 are less than 0.05 in all functions except F18. Moreover, the p-values of NVGOA1_2 and NVGOA2_2 are less than 0.05 in all functions except F14 and F18. For the NVGOA2_1 algorithm, p-values are less than 0.05 in F5, F9 and F23. Finally, p-values of the proposed NVGOA1_1 variant have been less than 0.05 in all functions except F4 and F8 which belong to the multimodal category, F13-F15, F17, and F18 which belong to the hybrid category, and F28 which belongs to the composition category.

Table 6 Results of Wilcoxon signed-rank test on CEC-2017 functions (case 1)
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From the previous explanation of the recorded results, it can be said that the best four variants from the proposed variants that have recorded the best results are NVGOA2_3 and NVGOA1_3, NVGOA3, and NVGOA4 compared to other proposed variants. But, all of the proposed variants have outperformed generally the original GOA except the NVGOA1_1 and NVGOA2_1 algorithms which have performance close enough to the performance of the original GOA.

These analyses illustrate the success of incorporating the mutation mechanism into GOA over the effect achieved by the grouping mechanism solely.

5.2.3 Case 2: comparative analysis with other state-of-the-art optimizers

In this section, the performance of the proposed NVGOA variants has been evaluated against other state-of-the-art optimization algorithms such as PSO, GWO, WOA, MFO, SSA, and HHO. The promising proposed variants as mentioned in the previous section have been chosen to conduct the comparative analysis. The chosen variants for comparison in this section are NVGOA1_3, NVGOA2_3, NVGOA3, and NVGOA4. These algorithms have been tested on CEC2017 benchmark functions. The parameter settings for the mentioned algorithms have been shown in Table 7. For fairness of the comparison, all chosen algorithms have been executed with the same initial populations for 30 dimensions while the size of populations is equal to 30, and the iterations number is set to 1000 as shown in Table 7. As well, each benchmark function has been executed for 30 independent runs.

Table 7 Parameter’s settings of the optimizers compared in case 2
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Table 8 summarizes the results of the conducted comparative analysis in case 2. The superiority of the tested NVGOA variants in most of the CEC-2017 functions has been shown as the proposed algorithms have gotten the best four overall ranks among the eleven algorithms tested as shown in Table 8.

Table 8 Simulation results of the chosen NVGOA variants compared to other optimizers tested on CEC-2017 benchmark functions (Case 2)
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However, the performance of the NVGOA algorithms in some unimodal functions (F1), multimodal functions (F5, F8, and F9), hybrid (F16, F17, F18, and F20), and some composition functions (F21, F22, F24, and F26) needs to be improved. The convergence curves of 12 functions are shown in Fig. 7. It is clear from Fig. 7 the proposed NVGOA variants have fast convergence well as better solutions in most functions among all other compared optimizers as clear in the performance of F1, F4, F6, F10, F11, F12, and F30 for instance.

Fig. 7
figure 7

Convergence curves of the promising NVGOA variants with other optimizers in case 2 (12 functions)

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Additionally, the Wilcoxon signed-rank test has been used to study the significant differences between the NVGOA2_3, the promising variant among the proposed NVGOA variants, and the other optimizers tested. The p-values of the Wilcoxon signed-rank test of case 2 are shown in Table 9. The p-value which is less than 0.05 in the table indicates that there is a significant difference between the two algorithms compared. As observed in Table 9, there are few values greater than 0.05 in PSO, SSA, GOA, GWO, and MFO columns. This observation has meant that the performance of these mentioned algorithms is different significantly from the performance of the NVGOA2_3 algorithm. Additionally, all the p-values of the WOA and HHO columns have less than 0.05. From these facts, it is deduced that the proposed NVGOA variants, especially the NVGOA2_3 algorithm, have better performance than the optimizers compared, which are the PSO, GWO, SSA, MFO, HHO, GOA, and WOA algorithms, in most of the CEC-2017 functions.

Table 9 The results of the Wilcoxon signed-rank test for algorithms compared in case 2
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5.2.4 Case 3: comparative study with previous enhancements of GOA

To further evaluate the performance of the proposed NVGOA variants, in this section, the NVGOA2_3, NVGOA1_3, and NVGOA4 variants' performance has been compared with previous enhancements of GOA on CEC-2017 benchmark functions. Those variants include SCFGOA (Saxena 2019), OLCGOA (Z. Xu et al. 2020a, b), AGOA (Wang et al. 2021), IGOA (Luo et al. 2018), and MOLGOA (Zhou et al. 2020).

It is noticed that the NVGOA3 in the two previous cases has results close to the NVGOA4, so just the performance of the NVGOA4 has been compared in this case.

Table 10 shows those variants’ results against the proposed NVGOA1_3, NVGOA2_3, and NVGOA4 algorithms on CEC-2017 benchmark functions. The overall rank and the average rank of all mentioned GOA variants have been calculated as well. By careful observation of Table 10, the proposed NVGO2_3, NVGOA1_3, and NVGOA4 enhancements have outperformed the performance of both IGOA and AGOA enhancements while OLCGOA has a little better performance compared to them. It is clear that the MOLGOA enhancement has the best performance in the chosen group of GOA enhancements including the suggested NVGOA.

Table 10 Comparison of results of the promising proposed NVGOA variants with other enhancements (Case 3)
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variants in this study, but it is undoubtedly that the NVGOA1_3, NVGOA2_3, and NVGOA4 have also good performance in some functions compared to MOLGOA.

Moreover, the performance of the best variant, which was developed by (Saxena 2019) and named SCFGOA7, has been compared with the proposed NVGOA1_3, NVGOA2_3, and NVGOA4 enhancements for 10 dimensions as well as 30 dimensions and 51 independent runs as shown in Table 11.

Table 11 Comparing the best enhancement results of (Saxena 2019) with the best enhancements of the NVGOA variants using both 10 and 30 dimensions for 51 independent runs (Case 3)
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By investigation of Table 11, the compared NVGOA variants have recorded the best solutions in all CEC-2017 functions in 10 dimensions comparison. As well, in the 30 dimensions comparison, the performance of the compared NVGOAs has been the best in all tested functions except F17, F18, and F24. This comparison study has confirmed the superiority of the best NVGOA variants against the developed SCFGOA7 enhancement for both 10 dimensions and 30 dimensions.

Finally, based on the results of those comparative cases, it is deduced that the combination of the two proposed mechanisms in the current study; the grouping mechanism for non-linear 'c' parameters and the mutation mechanism; have improved the GOA performance remarkably as well as sped up its convergence. In the next subsection, one of the real optimization problems has been stated. The NVGOA variants have been suggested to solve this optimization problem. The problem definition, case study, the simulation results, and a discussion of them will be shown in the next subsection.

5.3 Real application in smart grids

One of the important issues in the optimization sector is the balance between supply and demand in the smart electrical grid. A smart grid is a rising trend in the electric field that combines loads, storage systems, and distributed energy resources (DERs) into one controllable energy system (Green et al. 2013). The smart grid offers a powerful and distinctive feature called demand-side management (DSM) system that allows appliances to be responsive to price signals changing. The DSM refers to the efficient use of energy resources by changing a consumer's energy demand (Uddin et al. 2018). Electric utility companies use a demand response (DR) as a financial technique to force their clients to schedule their energy consumption into low-cost hours of the day. The DR proposes two types of strategies to minimize peak periods, operating cost, and the peak-to-average ratio (PAR): price-based techniques and incentive-based techniques (Asadinejad and Tomsovic 2017). In price-based techniques, the utility company can control the customers' devices indirectly via price signals sent. As a result of time-varying costs, customers are encouraged to make intelligent decisions such as scheduling their appliance operation at low-cost or off-peak hours (Mortaji et al. 2017). In incentive-based techniques, customers sign up a contract with the utility company to give it the option of remotely changing the states of appliances during high-peak periods or emergencies (Chen 2018). Consequently, an end-user can reduce his energy consumption and PAR through intelligently scheduling loads based on bi-directional communication between the smart meter and the electric utility company. If the appliance's scheduling is not intelligent, it may upset the customer's satisfaction. Wherefore, there is always a trade-off between reducing costs and waiting time (Khan et al. 2019).

In the current study, the proposed NVGOA variants have been applied to the load scheduling problem of three areas; commercial area, residential area, and industrial area; to decrease both the peak and the cost. The problem definition, the details of its equations, and the constraints will be shown and explained in the next subsection.

5.3.1 Problem Definition

Assume loadn is the forecasted load curve or the initial load curve. This curve must be changed to be close to the target curve. The expected electricity prices are inversely proportional to the target curve. After applying DSM, let the modified load curve obtained be.loadm. Then the main objective equation is (Gupta et al. 2016):

$${\text{Minimize}}\, f\,:\, \mathop \sum \limits_{t = 1}^{T} {\text{load}}_{m t} \, \times \,{\text{price}}_{t}$$
(21)

Equation (21) represents the total cost of electricity during T = 24 which is the number of time steps in a day, and the pricet refers to the price of electricity in each time step (t). To solve the minimization equation, a function ft is formed to obtain a modified load curve (loadm) as follows (Gupta et al. 2016):

$${\text{Minimize}}\, f_{ t } \,:\, \left| {LM} \right| - \left| {\Delta {\text{load}}} \right|$$
(22)

subject to, for all t = 1, 2… T. T is the total number of time steps. LM is a load margin, which is calculated for each t.

$$LM_{t} \, = \,{\text{Forecast}}_{t} - Tg_{t}$$
(23)

The above equation (Gupta et al. 2016) determines the marginal load that must be removed or added to the forecasted load curve at various hours to bring it as near to the objective curve as possible.

The target curve (\({Tg}_{t})\) can be formed by the following equation (Jamil and Mittal 2020):

$$Tg_{t} \, = \, \left( {\frac{{{\text{Price}} _{avg} }}{{{\text{Price}} _{max} }}\mathop \sum \limits_{t = 1}^{T} {\text{Forecast}} _{t} } \right)\, \times \, \frac{1}{{{\text{Price}} _{t} }}$$
(24)

where, T is the total number of time steps in a day i.e., 24 h, \({Price }_{avg}\) is the average price during the period T, \({Price}_{t}\) is the price of electricity at time step t, and \({Price}_{ max}\) is the maximum price through the day of T hours. The \({Forecast}_{t}\) is the value of day-ahead forecasted load consumption which has been collected from the data of the smart grid described in Table 12. Table 12 includes hourly forecasted load consumption of in three different areas: commercial, residential, and industrial as well as hourly wholesale electricity prices.

$$LM_{t} \, = \,\left\{ {\begin{array}{*{20}l} { \ge 0} \hfill & {{\text{Forecast}}_{t} \ge Tg_{t} } \hfill \\ { < 0} \hfill & {{\text{Forecast}}_{t} < Tg_{t} } \hfill \\ \end{array} } \right.$$
(25)
Table 12 Hourly forecasted load demand and hourly electricity prices (Logenthiran et al. 2012)
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The value of \({LM}_{t}\) given in the previous equation provides the information that whether \({\Delta load }_{t}\) calculated for a particular time step will be connected or disconnected as the following equation:

$$\Delta {\text{load }}_{t} \, = \,\left\{ {\begin{array}{*{20}l} {{\text{Load}}\_{\text{Disconn}}_{t} } \hfill & {{ }LM_{t} \ge 0} \hfill \\ {{\text{Load}}\_{\text{Conn}}_{t} } \hfill & {{ }LM_{t} < 0} \hfill \\ \end{array} } \right.$$
(26)

The above equation shows for any time step t, if the target load is less than the forecasted load means this time step t becomes an instant for disconnection, but if the target load is greater than the forecasted load means that time step t becomes an instant for connection. Thus, at each instant either connection or disconnection is done to obtain the modified load curve that is close to the target load curve.

\({Load\_Conn}_{t}\) is the total amount of load which needs to be connected in time step t. The amount of load to be connected at time step t (\({Load\_Conn}_{t}\)) is calculated through the following equation (Logenthiran et al. 2012):

$${\text{Load}}\_{\text{Conn}}_{t} \, = \, \mathop \sum \limits_{j = 1}^{A} N_{jt} P_{1j} \, + \,\mathop \sum \limits_{i = 1}^{l - 1} \mathop \sum \limits_{j = 1}^{A} N_{{j\left( {t + i} \right)}} P_{{\left( {1 + i} \right)j}}$$
(27)

The above equation calculates the amount of power consumption by The N number of appliances at the tth time step in the first summation part. If the appliance is having more than one operating time step then with the same quantity of appliances, power consumption is calculated at successive time steps, i.e., t + 1, t + 2… (t + l-1) in the second summation of the equation. \({N}_{jt}\) and \({N}_{j(t+i)}\) are appliances’ number of type j assumed to be operating at t and t + l. \({P}_{1j}\) and \({P}_{(1+i)j}\) are the quantities of power consumption of appliance j in the first-time step of the operation and at subsequent time steps respectively.

\({Load\_Disconn}_{t}\) is the total amount of load which needs to be disconnected from tth time step to load at the t time step gets decreased.? \({Load\_Disconn}_{t}\) is calculated through the following equation (Logenthiran et al. 2012):

\({\text{Load}}\_{\text{Disconn}}_{t} \, = \, \mathop \sum \limits_{j = 1}^{A} N_{jt} P_{1j} \, + \,\mathop \sum \limits_{i = 1}^{l - 1} \mathop \sum \limits_{j = 1}^{A} N_{{j\left( {t + i} \right)}} P_{{\left( {1 + i} \right)j}}\) (28_. this equation is similar to Eq. (27) in that all of the terms have the same meaning.

5.3.2 Constraints

The above minimization problem is constrained by the following constraints (Logenthiran et al. 2012):

1) Number of appliances of any type that are to be shifted will always be positive at any time step.

$$N_{jt} > 0\,\forall \,j, t$$
(29)

2) Number of appliances that are to be shifted at any t cannot be more than the available number of controllable appliances of that appliance type.

$$\mathop \sum \limits_{t = 1}^{T} N_{jt} \le {\text{Controllable}}\left( j \right)$$
(30)

where Controllable(j) is the total number of appliances of type j allowed for control.

5.3.3 Details of the controllable appliances in different smart grid areas

5.3.3.1 Residential area

The controllable appliances in the residential area usually have low power consumption ratings and little operating duration like the kettle, fan, dishwasher, etc. In the case study in this work, there are a total of 2604 controllable appliances of 14 different types. Table 13 shows the details of the controllable appliances in the residential area (Jamil and Mittal 2020; Logenthiran et al. 2012).

Table 13 Details of controllable appliances in the residential area
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5.3.3.2 Commercial area

The controllable appliances which are used in the commercial area usually have a higher power consumption rate than in the residential area. Table 14 shows the details of the controllable appliances in the commercial area There are about 800 controllable appliances of 8 different main types of appliances with their consumption ratings, starting time, and operating duration (Jamil and Mittal 2020; Logenthiran et al. 2012).

Table 14 Details of controllable appliances in the commercial area
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5.3.3.3 Industrial area

This area comprises the least number of controllable appliances that are characterized by large power consumption ratings than the appliances of another area and long operating duration. There are about 100 controllable appliances of 6 different types. Table 15 shows the details of the controllable appliances in the industrial area (Jamil and Mittal 2020; Logenthiran et al. 2012).

Table 15 Details of controllable appliances in the industrial area
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5.3.4 Simulation results for load scheduling problem and discussion

For more confirmation of the proposed NVGOA variants, the NVGOA algorithms have been proposed to solve the previous minimization problem given in Eqs. (21–30) against the original GOA. To deal with this problem, the day has been divided into time steps to apply the load-shifting mechanism. In this work, each time step has been equal to one hour, and the total number of time steps has been 24 h. The first-time step starts at 8:00 am. For applying the load scheduling technique, the appliances considered have an initial starting time provided by the user. The proposed NVGOA variants as well as the org GOA have been applied to the three mentioned areas which are residential, commercial, and industrial. The values of search space and maximum iterations in this experiment are 10 search agents and 100 iterations. All results in the figures and tables are the average values for 30 independent runs. Table 16 shows the results of the peak reduction percentage, cost reduction percentage, and peak-to-average ratio (PAR) in the residential area, commercial area, and industrial area. Also, the cost before and after applying the load-shifting mechanism using the org GOA and the proposed NVGOA variants has been calculated in addition to the peak load. As shown in Table 16, the org GOA and the proposed NVGOA algorithms have achieved a reduction in peak, cost, and PAR in comparison to the unscheduled load case. Although the org GOA has solved the defined problem with good results, the NVGOA algorithms outperformed it, and they have improved the ratio of cost reduction, peak reduction, and PAR more and more. The NVGOA1_1 has recorded the best reduction in the peak load in the residential area where the peak load has decreased from 1363.6 kW to 1037.7 kW with a 23.9% reduction, and PAR has been equal to 1.4. Also, the operating cost in the residential area has been recorded best reduction from 2302.9 $ to 2136.01$ with a 7.25% reduction by the NVGOA4 variant. The hourly load demand and hourly electricity cost obtained for the residential area before and after applying the proposed NVGOA variants-based load schedule technique are shown in Fig. 8 and Fig. 9.

Table 16 The peak reduction cost reduction and PAR before and after application of load shifting (LS) based NVGOAs
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Fig. 8
figure 8

The hourly load profile before and after application of load scheduling in the residential area

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Fig. 9
figure 9

The hourly cost of the residential area before and after the application of load scheduling

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Table 16 also shows the peak load reduction from 1818.2 kW to 1497.95 kW in the commercial area by all proposed NVGOA variants with a 17.61% reduction. Also, PAR has reduced from 1.7 to 1.4. The operating cost also has decreased from 3626.64 $ to 3292.88 $ by 9.2% with the application of the NVGOA3 variant. The hourly load demand and hourly electricity cost obtained for the commercial area before and after the application of the proposed variants are shown in Figs. 10, 11.

Fig. 10
figure 10

The hourly load profile before and after application of load scheduling in the commercial area

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Fig. 11
figure 11

The hourly commercial area before and after application of load scheduling

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As well, the peak load in the industrial area has been reduced from 2727.3 kW to 2476.48 kW by the proposed NVGOA2_1 with a 9.2% reduction. Also, PAR has reduced from 1.62 to 1.47 as the operating cost has reduced from 5712.05 $ to 4633.23 $ with an 18.9% reduction by applying the proposed NVGOA2_3 variant. The hourly load demand and hourly electricity cost obtained for the industrial area before and after the application of the proposed variants are shown in Figs. 12, 13. From these figures and Table 16, it is deduced that the proposed load-shifting strategy based on the suggested NVGOA variants has a great effect on the reduction of the peak demand, the utility cost, and PAR than the original GOA.

Fig. 12
figure 12

The hourly load profile before and after application of load scheduling in the industrial area

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Fig. 13
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The hourly industrial area before and after application of load scheduling

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6 Conclusion

In the presented work, the novel variants of GOA (NVGOA) have been proposed to mitigate the shortcomings of the original version of the grasshopper optimization algorithm (GOA) which is summarized in slow convergence which was leading to stuck in local solutions. In the proposed NVGOA variants, two proposed mechanisms; the grouping mechanism of non-linear ‘c’ parameters and the mutation mechanism; have been integrated into the GOA. Moreover, two different groups of non-linear ‘c’ parameters have been formed in the grouping mechanism. Comparative studies have been conducted on CEC-2017 benchmark functions for validation of the proposed novel variants of GOA (NVGOA). The numerical results have revealed that the proposed NVGOA variants have a significant effect to tackle the convergence drawbacks.

Firstly, the effectiveness of the proposed two mechanisms has been studied by comparing the eight proposed NVGOA algorithms' performance with the original GOA performance. The simulation results have illustrated that the NVGOA1_1 and NVGOA2_1 have performance close to the org GOA, but the other six proposed variants have exceeded the original GOA in performance. Besides that, the NVGOA1_3 has been faster in convergence than the NVGOA2_3 version, but the ability of the NVGOA2_3 to get the best solution has been higher. Along with this, the best four variants have been NVGOA2_3, NVGOA1_3, NVGOA3, and NVGOA4. Secondly, these four finest variants have been compared with different state-of-the-art optimizers, including WOA, PSO, GWO, MFO, HHO, and SSA. The comparison analysis has shown that NVGOA2_3, NVGOA4, NVGOA3, and NVGOA1_3 variants have outperformed the rest of their competitors. Also, the statistical analysis using the non-parametric Wilcoxon signed-rank test confirmed all that. Then, the promising variants of the proposed NVGOA algorithms have competed with other GOA enhancements presented in previous works including SCFGOA, OLCGOA, AGOA, IGOA, and MOLGOA. The comparison results have shown that the NVGOA2_3, NVGOA4, and NVGOA1_3 have ranked third to fifth ranks compared to OLCGOA, AGOA, IGOA, and MOLGOA. Accompanying, the three variants NVGOA2_3, NVGOA1_3, and NVGOA4 have surpassed the enhancement SCFGOA7 in both 10 and 30 dimensions. Finally, the NVGOA variants have been used to handle the load management problem in smart grids. The simulation results have exhibited that the application of the NVGOA variants to solve the load scheduling problem has been better than the original GOA in terms of peak demand reduction, operating cost reduction, and peak-to-average ratio (PAR) reduction. Where, the consumption at the peak has reduced by 23.9%, 17.6%, and 9.2% in residential areas, commercial areas, and industrial areas respectively. Also, the operating cost decreased by 7.25%, 9.2%, and 18.89% in residential areas, commercial areas, and industrial areas respectively. Therefore, the conclusion of the load scheduling problem has cleared that the NVGOA variants have been promising variants for dealing with real applications as well as benchmark problems. In future works, there are still many interesting topics for exploration. For instance, the presented NVGOA variants can be hybrid with other metaheuristic algorithms to improve their performance more. Furthermore, extending the proposed NVGOAs to multi-objective issues is also a topic worth discovering. Moreover, using the NVGOA variants as solvers of other real optimization problems is another interesting issue.